An X-ray imaging system, as shown in FIG. 1, may include an X-ray source 1 and a flat panel detector (FPD) 2 coupled to a gantry 3. Rotation of the gantry 3 causes the X-ray source 1 and the FPD 2 to move in a circular orbit around a patient positioned on a couch 4. The FPD 2 may receive X-rays emitted from the X-ray source 1 after the X-rays have passed through the patient. Because the received X-rays have been attenuated to various degrees by the patient's intervening tissues, the received X-rays may be used to generate a two-dimensional projection image of the tissues.
Two-dimensional projection images may be acquired from different gantry angles, and multiple two-dimensional projection images acquired from different gantry angles may be used collectively to build a three-dimensional reconstructed image of the patient anatomy. Accurate reconstruction of the three-dimensional image requires an accurate expression of a geometric relationship between the three-dimensional volume in which the imaging system resides (e.g., a treatment room) and the two-dimensional images acquired at a given gantry angle and FPD position.
Referring to FIG. 2, the rotation axis of the X-ray source is defined as the International Electrotechnical Commission (IEC) Yf axis. At the vertical gantry position (known as the zero degree position), the axis which passes through the X-ray source focal spot and which is also perpendicular to the IEC Yf axis is defined as IEC Zf axis. The axis perpendicular to the Y-Z plane that passes through the intersection of the Yf axis and Zf axis is defined as the IEC Xf axis. In an ideal trajectory, the X-ray source moves in a circular orbit in the vertical X-Z plane, and the center of this circle coincides with the origin Of of this IEC coordinate system. This point is also known as the machine isocenter. When the gantry is at the zero degree position, the IEC Zf axis points from the machine isocenter Of towards the X-ray source and the IEC Xf axis points towards the right while looking towards the gantry. With reference to FIG. 1, calibrated lasers 5 point to the machine isocenter Of and the crossed lines shown in FIG. 1 indicate axes in different directions.
FIG. 2 further illustrates a two-dimensional pixelized imaging coordinate system defined in the detector plane, defined by the U axis and the V axis. The U axis is the row axis parallel to IEC Xf, and the V axis is the column axis anti-parallel to the IEC Yf axis. The pixel coordinate (u=0, v=0) represents the top left corner pixel of the detector. The ideal detector plane is horizontal (perpendicular to the beam axis) and the planar imaging detector is “centered” by Zf axis when the gantry is at the zero degree (vertical) position. Moreover, the pixel coordinate of the center of the detector 2 is (W/2pu, H/2pv) where H (respectively pv) and W (respectively pu), respectively represent the width and height of the detector 2 (respectively width and height of each pixel). Also, the IEC coordinate of the point Of is ideally (0,0−(f−D)), where f and D represent “source to imager distance” (SID) and “source to axis distance” (SAD), respectively.
The actual trajectory of the X-ray source may differ from the above ideal description. For example, the X-ray source may move in and out of the vertical plane by a marginal amount. Moreover, as shown in FIG. 3, the trajectory of the X-ray source might not follow a perfect circle.
The relationship between the X-ray source and the FPD may also vary from the above-described ideal. For example, the relation between the detector assembly and the gantry might not be very rigid. Therefore, at each gantry angle, and due to gravity, the detector assembly may sag a different amount with respect to the central axis (CAX) of the beam (i.e., the line joining the X-ray source and the imaging isocenter), as shown in FIG. 4. In practice, the CAX meets the detector at a pixel location (u0, v0), which is somewhat different from (W/2pu, H/2pv). This point (W/2pu, H/2pv) is known as the principal point or the optical center. Also, as shown in FIG. 4, with respect to the ideal detector, the actual detector may exhibit (a) an out of plane rotation η about the axis u=u0 or (b) an out of plane rotation σ about the axis v=v0 or (c) a in plane rotation ϕ about the point (u0, v0).
However, it has been noted that the out of plane rotations η and σ are quite difficult to determine with reasonable accuracy and these two angles have only a small influence on the image quality compared to other parameters. In practical implementations, these two angles can be kept small (≤10) through good mechanical design and high-accuracy machining. It is therefore reasonable to assume that η=σ=0.
In summary, according to the above model, the plane of the detector is perpendicular to the CAX, but the trajectory of the X-ray source is not a perfect circle and the values of SID f and SAD D in FIG. 2 are not constant for all the gantry angles. Also, the two-dimensional coordinate system comprising the row (U) and column (V) vectors of the image exhibits in-plane rotation and translation about the CAX.
FIG. 5 illustrates these non-idealities. Without loss of generality and applicability to other X-ray imaging systems, the X-ray source of FIG. 5 is a linear accelerator equipped with beam collimation devices, including a multileaf collimator (MLC) that may rotate in a plane that is ideally parallel to that of the corresponding FPD and perpendicular to the CAX. Referring to FIG. 5, the image receptor coordinate axes X_r and Y_r are in the plane of the detector and aligned along X_bld and Y_bld respectively. Due to sag of the detector assembly, the principal point (u0, v0) may differ from the center of the panel. Also, the row (U) and column vectors (V) of the acquired image may exhibit rotation with respect to the projection of the coordinate axis of the MLC (at the zero degree position).
A projection matrix may be used to describe the geometric relationship between any three-dimensional point (xf, yf, zf) in the imaging room and its projection pixel coordinate (u, v) on the two-dimensional FPD. The projection matrix corresponds to a given position of the source/detector (i.e., X-ray source and FPD) pair. For an ideally circular source trajectory, the projection matrix may be written as:
      [                                        λ            ⁢                                                  ⁢            u                                                            λ            ⁢                                                  ⁢            v                                                λ                      ]    =                                                                                          [                                                                                                              1                          /                                                      p                            w                                                                                                                      0                                                                                              u                          0                                                                                                                                    0                                                                                                                          -                            1                                                    /                                                      p                            h                                                                                                                                                v                          0                                                                                                                                    0                                                                    0                                                                    1                                                                              ]                                θ                            ⋆                                                [                                                                                                              cos                          ⁢                                                                                                          ⁢                          ∅                                                                                                                                                  -                            sin                                                    ⁢                                                                                                          ⁢                          ∅                                                                                            0                                                                                                                                      sin                          ⁢                                                                                                          ⁢                          ∅                                                                                                                      cos                          ⁢                                                                                                          ⁢                          ∅                                                                                            0                                                                                                            0                                                                    0                                                                    1                                                                              ]                                θ                            ⋆                                                                                                            [                                                                                                              -                          f                                                                                            0                                                                    0                                                                    0                                                                                                            0                                                                                              -                          f                                                                                            0                                                                    0                                                                                                            0                                                                    0                                                                    0                                                                    1                                                                              ]                                θ                            ⋆                              [                                                                            1                                                              0                                                              0                                                              0                                                                                                  0                                                              1                                                              0                                                              0                                                                                                  0                                                              0                                                              1                                                                                      -                        D                                                                                                                        0                                                              0                                                              0                                                              1                                                                      ]                            ⋆                              [                                                                                                    cos                        ⁢                                                                                                  ⁢                        θ                                                                                    0                                                                                                                -                          sin                                                ⁢                                                                                                  ⁢                        θ                                                                                    0                                                                                                  0                                                              1                                                              0                                                              0                                                                                                                          sin                        ⁢                                                                                                  ⁢                        θ                                                                                    0                                                                                      cos                        ⁢                                                                                                  ⁢                        θ                                                                                    0                                                                                                  0                                                              0                                                              0                                                              1                                                                      ]                                                                ︷                  P          θ                      ⁢                                   ⋆                      [                                                                                x                    f                                                                                                                    y                    f                                                                                                                    z                    f                                                                                                1                                                      ]                          ,            where the symbol “*” denotes multiplication throughout this document.
However, as noted, the source trajectory is typically not a perfect circle of radius D in the vertical Xf Zf plane. [T]θ and [R]θ are the unknown translation and rotation, respectively, from the frame of reference of the aging/treat ent room to the frame of reference of the FPD. The subscript θ indicates that these transformations change with the gantry angle θ. Therefore, the projection matrix Pθ for a non-ideal source trajectory may be written as:
      [                                        λ            ⁢                                                  ⁢            u                                                            λ            ⁢                                                  ⁢            v                                                λ                      ]    =                                                                                          [                                                                                                              1                          /                                                      p                            w                                                                                                                      0                                                                                              u                          0                                                                                                                                    0                                                                                                                          -                            1                                                    /                                                      p                            h                                                                                                                                                v                          0                                                                                                                                    0                                                                    0                                                                    1                                                                              ]                                θ                            ⋆                                                [                                                                                                              cos                          ⁢                                                                                                          ⁢                          ∅                                                                                                                                                  -                            sin                                                    ⁢                                                                                                          ⁢                          ∅                                                                                            0                                                                                                                                      sin                          ⁢                                                                                                          ⁢                          ∅                                                                                                                      cos                          ⁢                                                                                                          ⁢                          ∅                                                                                            0                                                                                                            0                                                                    0                                                                    1                                                                              ]                                θ                            ⋆                                                                                                            [                                                                                                              -                          f                                                                                            0                                                                    0                                                                    0                                                                                                            0                                                                                              -                          f                                                                                            0                                                                    0                                                                                                            0                                                                    0                                                                    0                                                                    1                                                                              ]                                θ                            ⋆                                                [                  T                  ]                                θ                            ⋆                                                [                  R                  ]                                θ                                                                ︷                  P          θ                      ⋆                         [                                                            x                f                                                                                        y                f                                                                                        z                f                                                                        1                                      ]            
Calibration of an imaging system may include determination of the above projection matrix Pθ for the non-ideal source trajectory. Conventionally, this calibration involves imaging of a geometry calibration phantom. Geometry calibration phantoms typically consist of radio-opaque beads at known three-dimensional locations with respect to some known frame of reference. This reference frame may be within the phantom itself. The elements of the projection matrix are found by solving equations which relate the known three-dimensional locations within the phantom with the detected two-dimensional pixel locations in a projection image.
Others calibration methods that do not involve a phantom. For example, calibration may include an iterative reconstruction method of perturbing a model projection matrix and generating forward projections to best-match observed projections. Another method uses redundant projections over 180° and optimization to determine misalignment parameters with respect to an ideal projection matrix.
The existing methods for determining a non-ideal projection matrix are inefficient, time-consuming, based on inaccurate assumptions, and/or incompatible with treatment workflow.